In this Maple worksheet, we will consider evolutes. Since we will be plotting, we need...with(plots);

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Consider the curve parametrized by x1=t and y1=t^2. This is the parabola y=x^2. The equation of the normal lines is given by F1=(x-t)*(1)+(y-t^2)*(2t)=x-t+2yt-2t^3. Its envelope is given by R1=Resultant(F,diff(F,t)), which is R1=-216x^2-16+96y-192y^2+128y^3. To determine the envelope and obtain the plot of the result using Maple, we can use the following code. First, we declare the parametrization in t of x1 and y1.x1:=t;

NiM+SSN4MUc2IkkidEdGJQ==

y1:=t^2;

NiM+SSN5MUc2IiokSSJ0R0YlIiIj

We obtain the equation of the normal lines as follows, as well as calculating the resultant and plotting the result:F1:=simplify((x-x1)*diff(x1,t)+(y-y1)*diff(y1,t));

NiM+SSNGMUc2IiwqSSJ4R0YlIiIiSSJ0R0YlISIiKiZGKUYoSSJ5R0YlRigiIiMqJEYpIiIkISIj

r1:=resultant(F2, diff(F2,t),t);

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display({implicitplot(r1, x=-5...5, y=-5...5, scaling=constrained, numpoints=70000), implicitplot(y-x^2, x=-5...5, y=-5...5, scaling=constrained, numpoints=70000, color=blue)});

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In this example, let's consider the ellipse defined by the equation f=x^2+4y^2-4. First, we need to parametrize this ellipse. To this end, we will let t be the slope of the line connecting the point (x,y) on the ellipse to the point (-2,0). The equation of this line is then g=y-t(x+2). f:=x^2+4*y^2-4;

NiM+SSJmRzYiLCgqJEkieEdGJSIiIyIiIiokSSJ5R0YlRikiIiUhIiVGKg==

g:=y-t*(x+2);

NiM+SSJnRzYiLCZJInlHRiUiIiIqJkkidEdGJUYoLCZJInhHRiVGKCIiI0YoRighIiI=

Now we will use f and g to write x and y in terms of t. To obtain an equation for y, we will need to first eliminate x from f and g. To do this, we will use the resultant. Let h1=resultant(f,g,x)=4t^2y^2+y^2-4yt. Note that by Theorem 1, if we solve h1=0 for y, we will obtain an equation for each y-coordinate on the ellipse in terms of t.h1:=resultant(f,g,x);

NiM+SSNoMUc2IiwoKiZJInRHRiUiIiNJInlHRiVGKSIiJSokRipGKSIiIiomRihGLUYqRi0hIiU=

Solving, we get y=(4t)/(4t^2+1). We will define y2 to have this value so as to not duplicate variables.solve(h1,y);

NiQiIiEsJComSSJ0RzYiIiIiLCYqJEYmIiIjIiIlRihGKCEiIkYs

y2:=4*t/(1+4*t^2);

NiM+SSN5Mkc2IiwkKiZJInRHRiUiIiIsJiokRigiIiMiIiVGKUYpISIiRi0=

Similarly, to obtain an equation for x, let h2=resultant(f,g,y)=x^2-4+4t^2x^2+16t^2x+16t^2. Solving h2 for x, we obtain x=(-2(4t^2-1))/(4t^2+1), which we will define as x2. h2:=resultant(f,g,y);

NiM+SSNoMkc2IiwsKiRJInhHRiUiIiMiIiIhIiVGKiomSSJ0R0YlRilGKEYpIiIlKiZGLUYpRihGKiIjOyokRi1GKUYw

solve(h2,x);

NiQhIiMsJComLCYqJEkidEc2IiIiIyIiJSEiIiIiIkYtLCZGJ0YrRi1GLUYsRiM=

x2:=-2*(4*t^2-1)/(1+4*t^2);

NiM+SSN4Mkc2IiwkKiYsJiokSSJ0R0YlIiIjIiIlISIiIiIiRi4sJkYpRixGLkYuRi0hIiM=

Now we can obtain the equations for the family of normal lines using the following command and simplifying. Note: We multiply F by (1+4t^2)^3 to simplify since it does not affect the zero solutions of F, and we will redefine this simplification of F as G. F:=(x-x2)*diff(x2,t)+(y-y2)*diff(y2,t);

NiM+SSJGRzYiLCYqJiwmSSJ4R0YlIiIiKiYsJiokSSJ0R0YlIiIjIiIlISIiRipGKiwmRi1GMEYqRipGMUYvRiosJiomRi5GKkYyRjEhIzsqKEYsRipGMiEiI0YuRioiIztGKkYqKiYsJkkieUdGJUYqRjQhIiVGKiwmKiRGMkYxRjAqJkYuRi9GMkY3ISNLRipGKg==

G:=simplify((1+4*t^2)^3*F);

NiM+SSJHRzYiLC4qJkkieEdGJSIiIkkidEdGJSIiJCEkRyIqJkYqRilGKEYpISNLKiRGKkYrISQjPkYqIiNbKiZGKiIiJUkieUdGJUYpISNrRjRGMw==

Then we take the resultant to calculate the envelope of the normals.r:=resultant(G,diff(G,t),t);

NiM+SSJyRzYiLCQqJkkieUdGJSIiIiw2ISRIKEYpKiRJInhHRiUiIiMiJHMqKiRGKCIiJSEjRiokRigiIidGKSokRihGLiIkViMqJkYoRi5GLUYuIiRjKComRihGLkYtRjEiI1sqJkYoRjFGLUYuIiM3KiRGLUY0IiNrKiRGLUYxISRLJUYpIi1XcCF6KFtG

Finally, we graph the result.display({implicitplot(r, x=-3...3, y=-3...3, scaling=constrained, numpoints=70000), implicitplot(x^2+4*y^2-4, x=-3...3, y=-3...3, scaling=constrained, numpoints=70000, color=blue)});

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